The new House Republican health care bill to get rid of former President Barack Obama's Affordable Care Act and its tax on Americans who don't buy insurance coverage has a better way to incentivize participation, Senate Commerce Committee Chairman John Thune told CNBC on Tuesday.

Instead of Obamacare's individual mandate, the GOP's American Health Care Act, unveiled on Monday night, would allow insurers to charge a 30 percent penalty to consumers who have gaps in health coverage.

The third-ranking Senate Republican Thune said on "Squawk on the Street" the surcharge would only affect people who had coverage and dropped it and then want it again, whereas the Obamacare individual mandate penalizes those who don't get coverage in the first place.

The House GOP's plan does propose to keep the Obamacare provision protecting people with pre-existing conditions.

"People are going to have a choice when they get into a plan to stay in a plan," the South Dakota Republican Thune said, arguing the GOP approach discourages people from trying to "game the system" by dropping coverage and then signing back up again when they're sick.

The Republican plan also keeps another popular provision of Obamacare, which allows parents to cover young adults under 26 years old on their plans.

The House Ways and Means Committee and Energy and Commerce Committee are set to start the process of marking up the measure this week, even without a nonpartisan Congressional Budget Office analysis of the costs.

Mick Mulvaney, the new director of the Office of Management and Budget, told NBC's "TODAY" show Tuesday that "we will know what it costs before they vote on it." He predicted it's going to "save a dramatic amount of money" and be helpful to the deficit in the long term.

President Donald Trump is 100 percent, "all in" on the newly released Republican health-care bill, a White House official told CNBC earlier Tuesday.

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